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Banks' Stockholdings and the Correlation between Bonds and Stocks : A Portfolio Theoretic Approach Yoshiyuki Fukuda * [email protected] Kazutoshi Kan ** [email protected] Yoshihiko Sugihara *** No.13-E-6 March 2013 Bank of Japan 2-1-1 Nihonbashi-Hongokucho, Chuo-ku, Tokyo 103-0021, Japan * International Department (ex-Financial System and Bank Examination Department) ** Financial System and Bank Examination Department *** Personnel and Corporate Affairs Department (ex-Financial System and Bank Examination Department) Papers in the Bank of Japan Working Paper Series are circulated in order to stimulate discussion and comments. Views expressed are those of authors and do not necessarily reflect those of the Bank. If you have any comment or question on the working paper series, please contact each author. When making a copy or reproduction of the content for commercial purposes, please contact the Public Relations Department ([email protected]) at the Bank in advance to request permission. When making a copy or reproduction, the source, Bank of Japan Working Paper Series, should explicitly be credited. s Bank of Japan Working Paper Series

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Banks' Stockholdings and the

Correlation between Bonds and Stocks

: A Portfolio Theoretic Approach

Yoshiyuki Fukuda* [email protected]

Kazutoshi Kan** [email protected]

Yoshihiko Sugihara***

No.13-E-6 March 2013

Bank of Japan 2-1-1 Nihonbashi-Hongokucho, Chuo-ku, Tokyo 103-0021, Japan

* International Department (ex-Financial System and Bank Examination Department)

** Financial System and Bank Examination Department

*** Personnel and Corporate Affairs Department (ex-Financial System and Bank

Examination Department)

Papers in the Bank of Japan Working Paper Series are circulated in order to stimulate discussion and comments. Views expressed are those of authors and do not necessarily reflect those of the Bank. If you have any comment or question on the working paper series, please contact each author. When making a copy or reproduction of the content for commercial purposes, please contact the Public Relations Department ([email protected]) at the Bank in advance to request permission. When making a copy or reproduction, the source, Bank of Japan Working Paper Series, should explicitly be credited.

s

Bank of Japan Working Paper Series

1

BANKS’ STOCKHOLDINGS AND THE CORRELATION BETWEEN

BONDS AND STOCKS:*

A PORTFOLIO THEORETIC APPROACH

Yoshiyuki Fukuda†, Kazutoshi Kan‡ and Yoshihiko Sugihara§

ABSTRACT

In this paper, we analyze the optimal asset composition ratio of stocks and bonds for a bank

taking into consideration the correlation between the interest rate risk and equity risk in the

financial capital market using a portfolio model. The analysis reveals that in determining the

asset composition ratio in Japan, the correlation coefficient between the interest rate and stock

prices as well as the stock price volatility plays a more important role than the interest rate

volatility. We also show that in the present circumstances, the stockholding ratios of most

financial institutions in Japan are higher than the levels calculated from the model. It is

suggested that when the market is exposed to severe stress such as a surge in stock price

volatility or reversal of the correlation between the interest rate and stock prices, the

stockholding ratios would be even more excessive than the levels obtained from the model.

* The authors are grateful to Professor Alistair Milne (U.K. Loughborough University) and the

Bank of Japan’s staff for their useful comments in the process of developing this paper, and

Satomi Misawa and Minako Noutomi for their cooperation in its creation. Any errors are the

authors’ own. The views expressed here are those of the authors and do not reflect the official

views of the Bank of Japan or its Financial System and Bank Examination Department.

†International Department (ex-Financial System and Bank Examination Department)

([email protected])

‡Financial System and Bank Examination Department ([email protected])

§Personnel and Corporate Affairs Department (ex-Financial System and Bank Examination

Department)

2

1.Introduction

While a bank earns profits by holding a variety of assets including bonds, stocks and loans, it

is also exposed to the risk associated with price fluctuations of these assets. It is therefore

necessary to pay particular attention to the risk management in stockholding by the Japanese

banks. Though the banks reduced their stockholdings through the 2000s, the outstanding

amount of stocks held by financial institutions remains high and the equity risk represents a

significant risk factor for bank management (Figures 1 and 2).

In Japan since the 2000s, the interest rate risk and the equity risk have been negatively

correlated, albeit slightly, so banks are believed to have enjoyed a risk hedging effect by

simultaneously holding stocks and bonds (Figure 3). That is, in most cases, when the interest

rate rose, unrealized losses on bonds were set off by unrealized gains on stocks. As such risk

hedging effect is produced between asset classes, the loss of the overall portfolio can be

reduced to a lower level than would be expected from individual risks. On the other hand, in

Japan in the 1990s after the bursting of the bubble economy and in Italy around the end of

2011 when concerns grew about the sovereign risk, there were times when stock prices

declined significantly while bond interest rates rose at the same time (i.e., simultaneous

declines in stock and bond prices) (Figure 4). In such a case, the loss would be greater than

expected because the interest rate risk would be positively correlated with the equity risk and

both asset classes would produce losses. Such cases indicate that the total risk faced by a

bank may be over- or under-estimated if the risk of each asset class is separately evaluated

and simply summed up. Therefore, banks face a difficult problem of determining the asset

composition so as to maximize profits while striking a balance among the risk of different

asset classes under capital constraints.

There have been many studies on the asset composition (portfolio) optimization problem,

including the study by Markowitz (1959). Among others, studies which analyzed a mixed

portfolio of bonds and stocks include those by Konno and Kobayashi (1997) and Fischer and

Roehrl (2003). Konno and Kobayashi (1997) derived the optimal allocation for a large-scale

portfolio comprised of bonds and stocks through simulation. Fischer and Roehrl (2003), when

they posed a portfolio selection problem, considered a number of optimization criteria

including the expected shortfall and RORAC and performed a comparative analysis of their

3

performances through simulation1.

The purpose of this paper is to obtain the optimal asset composition ratio of bonds and stocks

based on the portfolio model for a bank when the bank faces a capital constraint. The

correlation between the equity risk and the interest rate risk in Japan is expressly taken into

consideration. This paper also analyzes how the asset composition ratio changes in response

to stress imposed on the market environment. When the market is in times of stress such as a

financial crisis, for example, the hedging effect of stock and bond holdings may be lost due to

a sharp decline in stock prices and the reversal of the correlation between the interest rate

and stock prices. In such a case, the optimal asset composition ratio also changes.

This paper is organized into five sections. Section 2 describes the model of the optimization

behavior of the banks' bond and stockholding ratios. The correlation between bonds and

stocks is expressly considered. Section 3 shows the qualitative results obtained from the

estimated model. Section 4 compares the data and the results obtained from the model and

attempts to assess whether the stockholding ratio of a financial institution is appropriate

relative to the capital buffer. Section 5 contains the conclusions and future issues.

1 RAROC (Risk Adjusted Return On Capital) is an indicator which expresses the earning strength relative

to the allocated capital, calculated as “profits after expenses/economic capital”. “Economic capital” here may refer to the amount of risk or to the allowable maximum amount of risk.

4

2.Analytical Method

2.1 Model

The analysis assumes a portfolio model comprised of two financial instruments: stocks and

bonds2. Of these, for the bond portfolio, the bank is assumed to trade bonds with duration 𝐷.

The bank is also assumed to determine the holding ratios of stocks and bonds so as to

maximize the expected return of portfolio while controlling the total amount of risk of the

portfolio containing the equity risk and interest rate risk at or below the allowable level

relative to the capital constraint.

If the present time is 0 and the investment duration is 𝑇, the bank’s investment strategy

can be expressed as3:

max{𝒘} 𝐄0 𝑤1 𝐵𝑇 ,𝐷+𝑐0,𝐷𝑇

𝐵0,𝐷− 1 + 𝑤2

𝑆𝑇

𝑆0− 1 , (1)

s.t.

𝐕0 𝜙𝛵 ≤ 𝛾, 𝛾 > 0 (2)

𝑤1 + 𝑤2 = 1 3

where, 𝐄0[] and𝐕0[] represent the expectation and the variance at time 0 respectively. 𝐵𝑇,𝐷

and 𝑆𝑇 represent the value of the bond and stock portfolio at time 𝑇, respectively. 𝜙𝑇 ≡

𝑤1 (𝐵𝑇,𝐷 + 𝑐0,𝐷𝑇)/𝐵0,𝐷 − 1 + 𝑤2 𝑆𝑇/𝑆0 − 1 is the rate of return from the portfolio from

time 0 to 𝑇, 𝒘 = 𝑤1,𝑤2 is the investment weight on bonds/stocks, 𝑐0,𝐷 is the coupon

amount of the bond at maturity 𝑐0,𝐷4, and 𝛾 is the variance (volatility) of the allowable

2 There are three major types of a bank’s key asset holdings: loans, bonds and stocks. The analysis

in this paper assumes that banks determine the loans first, and on that basis, address the problem

of how to allocate the remaining assets to bonds and stocks.

3 In setting the problem, the constraint condition 𝑤1,𝑤2 ≥ 0 is not imposed. That is, the

assessment is made on the basis that short positions can be taken in stocks and bonds. However,

even when such non-negative constraint is imposed, there is no essential difference in setting

between them since their solutions conform to each other while the condition 𝑤1 ,𝑤2 ≥ 0 is

satisfied.

4 In setting the problem, the variance of the securities portfolio was used as amount of risk;

however, actually, banks are supposed to perform risk assessment in reference to the quantiles in

the loss distribution such as Value at Risk. However, since the variance of the portfolio is

proportional to the quantiles when the changes in bond prices and stock prices follow the

5

maximum portfolio for a bank which is given exogenously. The value of the stock portfolio

includes dividends. In contrast, the coupon of the bond portfolio is paid as a lump sum on

the last day of the investment period.

Here, 𝐵𝑡,𝐷 and 𝑆𝑡 , which are the portfolio values of the bonds and the stocks, respectively,

are assumed to take the following stochastic processes:

𝑑𝑆𝑡 = 𝜇𝑆𝑡𝑑𝑡 + 𝜍𝑠𝑆𝑡𝑑𝑊𝑡 1 (4)

𝑑𝑟𝑡 = 𝜅 𝜃 − 𝑟𝑡 𝑑𝑡 + 𝜍𝑟𝑑𝑊𝑡 2 (5)

𝑑𝐵𝑡,𝐷 = −𝐷 ∙ 𝐵𝑡 ,𝐷𝑑𝑟𝑡 (6)

𝑑𝑊𝑡 1 ∙ 𝑑𝑊𝑡

2 = 𝜌𝑑𝑡 (7)

where, 𝑊𝑡(1)

and 𝑊𝑡(2)

represent Brownian motion, 𝑟𝑡 represents the interest rate during the

period to maturity 𝐷 (𝐷 > 0), 𝜇 represents the expected rate of return of the stocks

including dividends, 𝜍𝑠 represents the volatility of the stock portfolio, 𝜅 represents the

mean reversion speed of the interest rate, 𝜃 represents the mean reversion level of the

interest rate, 𝜍𝑟 represents the interest rate volatility of the bond portfolio, and 𝜌 represents

the correlation coefficient of the Brownian motion. As shown in equation (6), for changes in

the value of the bond portfolio, only the effect of the parallel shift in the yield curve is taken

into consideration (the effect of changes in the slope or curvature of the yield curve is not

considered)5, 6

. Equation (5) was developed as a model to express fluctuations in the spot

normal distribution, the problem setting by equation (1) can be considered to simulate a bank’s

actual behavior.

5 If 𝐵𝑡 ,𝐷 is a function of 𝑟𝑡 , according to Ito’s lemma, 𝑑𝐵𝑡 ,𝐷 = −𝐷 ∙ 𝐵𝑡 ,𝐷𝑑𝑟𝑡 +

1

2

𝜕2𝐵𝑡 ,𝐷

𝜕𝑟𝑡2 𝜍𝑟

2𝑑𝑡. If we

ignore the effect of second- and higher-order changes in the interest rate on the bond price,

since 𝜕2𝐵𝑡 ,𝐷/𝜕𝑟𝑡2 = 0, we obtain equation (6). If the interest rate changes significantly, the

effect of second- and higher-order changes may not be ignored, but here, only the first-order

changes are considered for convenience.

6 Equation (6) only considers the parallel shift in the yield curve because when key rates in the

number of 𝑟𝑖 𝑖=1𝐼 and the key rate duration 𝐷𝑖 𝑖=1

𝐼 are considered, and if the bank invests 𝜔𝑖

only in bonds with the relevant duration, the average duration of the bond portfolio 𝐷 is

𝐷 = 𝐷𝑖𝐼𝑖=1 ∙ 𝜔𝑖/ 𝜔𝑖

𝐼𝑖=1 . On the other hand, if we ignore the effect of second- and higher-order

changes in the interest rates on the bond price, the change in the value of the bond portfolio

𝐵 = 𝜔𝑖𝐼𝑖=1 is 𝑑𝐵 = − 𝐷𝑖

𝐼𝑖=1 ∙ 𝜔𝑖𝑑𝑟𝑖 . Therefore, if the yield curve makes a parallel shift, since

6

rates. Since the bond’s duration is fixed in the analysis performed in this paper, the

fluctuations in the government bond interest rates are expressed by such spot rate model7.

2.2 Optimal asset holding ratio

Subject to the above assumptions, the solutions of equations (1) to (3) are as follows:

𝑤1 =

𝑐−𝑏− 𝑏2−𝑎𝑐+𝛾 𝑎−2𝑏+𝑐

𝑎−2𝑏+𝑐

𝑤2 =𝑎−𝑏+ 𝑏2−𝑎𝑐+𝛾 𝑎−2𝑏+𝑐

𝑎−2𝑏+𝑐

(8)

However,

𝑎 = 𝐕0 𝐵𝑇,𝐷/𝐵0,𝐷 = 𝑋2𝑌2 𝑌2 − 1 (9)

𝑏 = 𝐂𝐨𝐯0 𝐵𝑇,𝐷/𝐵0,𝐷 , 𝑆𝑇/𝑆0

= 𝑋𝑌𝑒𝜇𝑇 exp −𝐷𝜌𝜍𝑠 𝜍𝑟

𝜅 1 − 𝑒−𝜅𝑇 − 1 (10)

𝑐 = 𝐕0 𝑆𝑇/𝑆0 = 𝑒2𝜇𝑇 𝑒𝜍𝑠2𝑇 − 1 (11)

where,

𝑋 = exp −𝐷 𝑒−𝜅𝑇 − 1 𝑟0 − 𝜃 −1

2𝐷2𝜍𝑟

2𝑇 (12)

𝑌 = exp 𝐷2𝜍𝑟

2

4𝜅 1 − 𝑒−2𝜅𝑇 (13)

(see the addendum for details of the derivation process). Therefore, if the parameters are

determined, the optimal stockholding ratio can be uniquely elicited analytically by

determining 𝛾 (the variance of the portfolio) from equation (8)8.

𝑑𝑟𝑖 = 𝑑𝑟 𝑖∀ , 𝑑𝐵 = −𝐷 ∙ 𝐵𝑑𝑟 and thus the relationship in equation (6) is established.

7 Given the movement in the interest rates shown in Figure 3, as mean reversion behaviors have

been generally observed since 2000, it is considered to be appropriate to express the interest rate

in the mean reversion process such as equation (5).

8The optimal holding ratio becomes an imaginary number and may not be elicited depending on

the level of 𝛾 or volatility. This occurs when the amount of risk associated with the portfolio

exceeds 𝛾 regardless of the setting of the stockholding ratio because the amount of existing bond

holdings is too large compared to the volatility.

7

2.3 Parameters

Parameters are set using calibration and the maximum likelihood estimation. For particular

parameters (𝜍𝑠, 𝜍𝑟 , and 𝜌), the values are set for two cases: the benchmark estimated by the

samples since early 2000 and when the financial market is under stress9. In this way, it is

possible to assess how the stockholding ratio changes in times of stress.

a. Calibration

Some parameters are set exogenously based on data, etc. The bank’s investment period 𝑇

is exogenously set at 1 year (𝑇 = 1 year). The duration of the bond portfolio 𝐷 is set at 2.6

years and 3.9 years, respectively, for major banks and for regional banks based on the actual

figures as of the end of March 2011. The coupon level of the bond portfolio is set so that it

corresponds to the interest rate level of the currently issued bond whose maturity is close to

the duration (𝑐0,𝐷 = 𝐵0,𝐷𝑟0). The expected rate of return of the stock portfolio is the

average figure for the last 30 years (𝜇 = 7.77%)10

. For the correlation function 𝜌, the average

correlation function estimated based on stock returns and the changes in interest rates since

early 2000, 0.33, is used as the benchmark parameter. For the correlation function, the

correlation coefficient at the time when stock prices and interest rates declined

simultaneously after the bursting of the economic bubble, −0.63, is used as the parameter in

times of stress (Figure 3).

b. Parameter estimation

First, from equation (5), the bond portfolio is assumed to follow the normal distribution

where the mean and the variance of the interest rate are expressed as follows:

𝐄0 𝑟𝑡 = 𝑒−𝜅𝑡 𝑟0 − 𝜃 + 𝜃 14

𝐕0 𝑟𝑡 =𝜍𝑟

2

2𝜅 1 − 𝑒−2𝜅𝑡 15

9The effect of 𝜇, 𝜅,and 𝜃 on the stockholding ratio is not much different between the benchmark

and in times of stress compared to the effect of the volatility and the correlation.

10 Since the average stock return calculated for the past 10 years takes a negative value, the

solution of the optimization fails to describe the bank’s realistic investment behavior. This analysis

adopted a long-term average for the last 30 years given that in fact, the stock return assumed by

investors should be higher than this level.

8

If the number of samples is 𝑁, the log likelihood of the bond portfolio 𝐿𝐵 is expressed as:

𝐿𝐵 = −𝑁

2ln 2𝜋 −

𝑁

2ln

𝜍𝑟2

2𝜅 1 − 𝑒−2𝜅𝜏 −

𝜅

𝜍𝑟2 1 − 𝑒−2𝜅𝜏 𝑟𝑖𝜏 − 𝑒−𝜅𝜏 𝑟0 − 𝜃 + 𝜃 2

𝑁

𝑖=1

(16)

Here, is the sample time interval and is set as 𝜏 = 1/250. 𝜅, 𝜃, and 𝜍𝑟 are estimated to

be the parameters which maximize 𝐿𝐵 .

The stock portfolio is assumed to follow the log-normal distribution where the mean and

the variance are expressed as follows from equation (4):

𝐄0 ln 𝑆𝑡𝑆0 = 𝜇𝑡 −

𝜍𝑠2𝑡

2 17

𝐕0 ln 𝑆𝑡𝑆0 = 𝜍𝑠

2𝑡 18

The log likelihood of the stock portfolio 𝐿𝑆 is expressed as:

𝐿𝑆 = −𝑁

2ln 2𝜋 −

𝑁

2 ln (𝜍𝑠

2𝜏) −1

2𝜍𝑠2𝜏 ln

𝑆𝑖𝜏𝑆𝑖𝜏−1

− 𝜇𝜏 −𝜍𝑠

2𝜏

2

2

(19)

𝑁

𝑖=1

Here, the partial differential value for 𝜍𝑠 and 𝜇 is zero:

𝜇 =1

𝑁𝜏 ln

𝑆𝑖𝜏𝑆𝑖𝜏−1

+ 𝜍𝑠

2

2 (20)

𝑁

𝑖=1

𝜍𝑠2 =

1

𝑁𝜏 ln

𝑆𝑖𝜏𝑆𝑖𝜏−1

−1

𝑁 ln

𝑆𝑖𝜏𝑆𝑖𝜏−1

𝑁

𝑖=1

2

(21)

𝑁

𝑖=1

(see Table 1 for basic statistics).

The estimation uses the composite index of the interest rates of currently issued 3-year

bonds calculated by Bloomberg as the data for the interest rate 𝑟𝑡 (see Table 1 for the basic

statistics of the samples used for the estimation). TOPIX is used as the market portfolio data

of stock prices.

Of the parameters, 𝜃, 𝜅, 𝜍𝑠 and 𝜍𝑟 are estimated based on the daily data from early 2000

9

to the end of November 2011. In particular, for 𝜍𝑠 , 𝜍𝑟 and 𝜌 parameters are estimated for

two cases: the benchmark and the times of stress. The estimation results based on all samples

are used as the estimates for the benchmark parameters. For the parameters for the times of

stress, estimation is performed with a 1-year rolling window and the estimation results at the

time when volatilities 𝜍𝑠 and 𝜍𝑟 are the greatest and at the time when the correlation

coefficient 𝜌 is the smallest are used as the respective estimates.

Table 2 shows the estimation results. The mean reversion level of the interest rate 𝜃 and

the mean reversion speed of the interest rate 𝜅 are estimated to be 0.45% and 0.52,

respectively. The stock price volatility 𝜍𝑠 is estimated to be 23.1% for the benchmark. For the

parameters in times of stress, the stock price volatility increases to 42.4% at the time of the

Lehman Shock. The interest rate volatility 𝜍𝑟 is estimated to be 0.30% for the benchmark and

the estimate rises to 0.49% at the time of the Lehman Shock.

3.Assessment for the Whole Banking Sector

3.1 The stockholding ratio and the variance of the benchmark

Figure 5 shows the relation between the stockholding ratio and the variance of the total

portfolio of major and regional banks under the benchmark parameters. The vertical and

horizontal axes represent the variance 𝛾 and the optimal stockholding ratio 𝑤2∗,

respectively.

From the results, firstly, we can see that there is a non-linearity between the stockholding

ratio and the variance. This is because the variance 𝛾 and the stockholding ratio 𝑤2∗

obtained analytically in equation (8) have a binominal series relation. As the stockholding

ratio rises, the variance of the bank’s portfolio becomes larger at an accelerated pace.

Secondly, since the average duration of bond investments is longer and the amount of risk

associated with bond holding is larger for regional banks, if the stockholding ratio is low, the

variance of regional banks’ portfolios is larger than that of major banks. Thirdly, when the

stockholding ratio rises to about 10%, the difference in the portfolio variance between major

and regional banks is reduced. As the stockholding increases, the bondholding decreases

because the amount of equity capital which can be allocated to the amount of interest rate

risk associated with the bondholding is reduced. Therefore, for regional banks in particular,

10

the variance associated with bond investment is more significantly reduced. At present (the

end of the first half of FY2011), since the stockholding ratios of major banks are higher than

those of regional banks, the amount of risk of the securities portfolio is larger for major banks.

3.2 Portfolio variance and the amount of risk in financial institutions

So far, we have explored the relation between the stockholding ratio and the variance of

the total securities portfolio of financial institutions. Incidentally, if we use the portfolio

variance value, we should be able to calculate the amount of loss suffered by a financial

institution based on certain assumptions, which is typically VaR. If we assume that the rate of

return of the portfolio follows the normal distribution, the amount of loss which may be

suffered by a financial institution with a fixed probability can be calculated by giving the

variance value to VaR. This is also the amount of risk held by the financial institution.

Specifically, this can be calculated by multiplying the volatility by a constant and then

multiplying by the portfolio value. Here, since 99%VaR is assumed as the amount of risk, it is

multiplied by 2.33, which is 1 percentile of one side of the normal distribution. Based on

these assumptions, subject to the current stockholding ratio, major and regional banks would

suffer losses exceeding 5 trillion yen and 2 trillion yen, respectively, with a probability of 1%

(the vertical line in the figure represents the stockholding ratio as of the end of the first half of

FY 2011).

3.3 Effect of different financial market conditions on the stockholding ratio and

the amount of risk

This section analyzes the effect of different financial market conditions on the stockholding

ratio and the amount of risk. Here, three situations are assumed as the financial market

conditions: where the correlation coefficient between the interest rate and the stock return 𝜌

declines, where the stock price volatility 𝜍𝑠 increases, and where the interest rate volatility

𝜍𝑟 increases. Figures 7 to 12 show the total amount of risk given the stockholding ratio at the

present time when the correlation coefficient between the interest rate and the stock return,

the stock price volatility and the interest rate volatility are changed continuously.

Figures 7 and 8 show the amount of risk for different correlation coefficients. When the

11

correlation coefficient moves in a negative direction, since the effect of holding stocks to

hedge the bond losses is lost and losses arise simultaneously from the stocks and bonds, the

amount of risk should increase. For example, since the hedging effect is lost when the

correlation coefficient becomes zero, the amount of risk increases by approx. 0.6 trillion yen

and 0.5 trillion yen for major and regional banks, respectively, compared to the benchmark

case. On the other hand, if the correlation coefficient reverses to −0.63 as seen in times of

crisis, since the stock and bond portfolios move in the same direction, the amount of risk

increases by approx. 1.6 trillion yen and 1.2 trillion yen for major and regional banks,

respectively compared to the benchmark case.

Figures 9 and 10 show the amount of risk when the stock price volatility rises. The higher

the volatility, the larger the amount of risk associated with stockholding would become, thus

increasing the total amount of risk. For example, if the stock price volatility rises to the level

at the time of the Lehman Shock, the amount of risk increases by approx. 5 trillion yen and 2

trillion yen for major and regional banks, respectively, compared to the benchmark case.

Figures 11 and 12 show the amount of risk when the interest rate volatility rises. Since the

higher the volatility, the larger the amount of risk associated with bonds, the total amount of

risk is increased. However, the size is smaller compared to the case of stock price volatility. If

the interest rate volatility rises to the level at the time of the Lehman Shock under stress, the

amount of risk increases by approx. 0.1 trillion yen and 0.5 trillion yen for major and regional

banks, respectively, compared to the benchmark case.

From the above results, we can see that in Japan, the correlation coefficient between the

interest rate and stock prices and the stock price volatility play more important roles than the

interest rate volatility in determining the bank’s asset composition ratio. This implies that

banks should reduce their stockholdings because of their large amount of risk.

4.Assessment for Individual Financial Institutions

4.1 Assessment of the benchmark

This section compares the stockholding ratio calculated from the model and the actual

stockholding ratio of individual financial institutions according to their equity capital

requirements. Here, the amount of equity capital which can be allocated to bonds and stocks

12

(= capital buffer) is defined as Tier1 capital less the amount of regulatory equity capital and

the credit risk, the operational risk and the foreign bond holding risk11, 12

. This is because core

capital is considered to be primarily used to absorb the risk associated with lending, which is

banks’ main line of business. The amount of the foreign bond holding risk was subtracted in

advance since such risk is not considered in this model.

In Figure 13, the horizontal axis represents the difference between the allowable

stockholding ratio according to the capital buffer calculated from the model and the actual

stockholding ratio and the vertical axis represents the number of banks. The banks showing

negative figures hold more stocks than the level allowed by the capital buffer. Overall, about

20% of the banks hold excessive stocks relative to the capital buffer. In addition, both among

major and regional banks, some banks’ stockholding ratios are quite high and certain banks’

stockholding ratios are excessive and exceed 5% of the capital buffer.

These results show that even in case of the benchmark, some banks hold greater equity risk

than the allowable amount of risk and such banks need to further reduce their stockholdings.

11

The amount of regulatory equity capital is set at 8% and 4% of the risk assets for international

and domestic reference banks, respectively. The amount of the credit risk is assumed to be the

non-expected loss at the confidence level of 99%, and is estimated based on the default

probability calculated from the borrower classification data for bank lending and the collection

rate of bank loans in the event of loss. The amount of the foreign bond holding risk was calculated

by multiplying VaR of foreign interest rates at the confidence level of 99% by the average maturity

of bonds. Here, VaR of foreign interest rates was obtained by weighted-averaging VaR of 3-year

government bond interest rates in each currency based on the Japanese banks’ outstanding bond balances by country estimated by the balance of payments statistics. The amount of operational is set at 15% of the gross profit. 12

This paper subtracted the amount of the regulatory equity capital requirement in calculating the capital buffer to be allocated to bonds and stocks, taking the bank’s reputational risk into consideration. Should heavy losses arise from a bank’s securities portfolio and the bank’s equity capital ratio falls below the regulatory requirement level, prompt corrective action is implemented on the bank by the supervisory authorities. If, as a result, credit uncertainty spreads about the bank, it may trigger a situation such as a run on the bank, illustrating the bank’s reputational risk. Therefore, though the regulatory equity capital in essence is the stock of loss absorption potential to provide for the occurrence of a tail risk, in reality, it is important not to fall below the regulatory standard even in time of crisis involving significant deterioration of the financial environment.

13

4.2 Effect of different financial market conditions

Next we compare the stockholding ratio allowed based on the capital buffer and the actual

stockholding ratio when the financial market conditions change and the correlation

coefficient between the interest rate and stock return 𝜌 declines, the stock price volatility 𝜍𝑠

rises and the interest rate volatility 𝜍𝑟 rises compared to the benchmark conditions.

Figures 14, 15 and 16 show the difference between the allowable stockholding ratio subject

to the capital buffer based on each parameter and the actual stockholding ratio for the

correlation coefficient, the stock price volatility and the interest rate volatility, respectively. In

all cases, more banks hold excessive stocks relative to the capital buffer compared to the

benchmark case (Figure 13). This result implies that both major and regional banks need to

reduce their stockholdings significantly given the possibility of market environment

deterioration. In particular, for major banks, the stockholding ratio of most of the banks

becomes excessive in the scenario where the stock price volatility increases, reflecting the fact

that the near-term level of the stockholdings is high. For regional banks, the stockholding

ratio of most of the banks becomes excessive in the scenario where the correlation coefficient

between the interest rate and stock prices is reversed, reflecting the fact that the durations of

bonds are longer than those held by major banks.

4.3 Issues to be considered

There are several issues to be considered about the analysis results so far in relation to the

assumptions. First is the amount of capital which serves as a constraint. In the above

calculation, we assumed that the amount of equity capital less the equity capital to cover the

lending risk and the regulatory equity capital is allocated to the securities portfolio. Under

such assumption, the stockholding ratio is calculated conservatively. If the regulatory equity

capital is allocated to the amount of risk of the securities portfolio without subtraction, as a

matter of course, the ratio of stocks held rises as well. However, if the losses from the

securities portfolio actually erodes the regulatory equity capital due, for example, to a stock

price decline, this may cause difficulties in normal business operations as a result of

deterioration of the reputation in the market and financing problems. In the above calculation,

we used a conservative assumption for the amount of risk allocated to securities taking these

14

factors into consideration.

The second is the relationship with the financial institution’s risk management. When a

financial institution controls its risk, it first calculates the potential losses incurred at the time

of stress and considers whether the sum total of such losses remains within the range of the

capital. For the purpose of the above analysis, this is the same as calculating the amount of

risk based on the assumptions that the correlation between asset classes is the highest and

that stocks and bonds suffer maximum losses at the same time. Furthermore, the amount of

risk is assessed inclusive of the regulatory capital. Accordingly, as mentioned earlier, the

analysis in this paper is different from the risk management method of an ordinary financial

institution in that it assumes conservative capital use as it first subtracts the amount of

regulatory capital and then measures the level of stocks available for holding relative to the

remaining capital.

The third is the assumption about the stock return. The analysis in this paper used the

average value over the last 30 years (7.77%) as the stock return. However, if we calculate the

latest return (over the last 10 years), the average value becomes negative and the expected

dividend rate (as of the end of March 2011) is also at a quite low level of 1.66%. Therefore, if

we use the latest return, the stockholding ratio becomes even lower. On the other hand, it is

also important how to take into consideration the advantages from other banking activities

obtained as strategic stockholdings in addition to direct returns. Some consider that the

benefit of earning commission income and loan margins from a company stably for a long

time by holding the company’s stock for a long time should be added to the direct return

obtained from the stocks. If such return is actually measurable, it would be possible to add

such return to the direct return to calculate the total return, which would then be used to

work out the optimal stockholding ratio.

5.Conclusions and Future Issues

This paper calculated the optimal asset composition ratio using the portfolio model taking

into consideration the correlation between the equity risk and the interest rate risk. Within

the range of variations in the parameters since the 1990s, it was identified that fluctuations in

the stock price volatility and the correlation coefficient of stock prices and the interest rate

15

had more impact on the stockholding ratio than the interest rate volatility. We also found that

some banks held more stocks than the allowable level based on the capital buffer. We

concluded that, when the financial market is under stress, the percentage of such banks will

increase.

Lastly, we suggest two directions to further develop this paper. The first is to analyze the

lending, which is a bank’s core business, with consideration of the correlation with other

asset classes. The credit risk has the same distribution of losses and profits as stocks and

bonds and needs to be analyzed together with the other two types of risks. The second issue

is the necessity of expanding the analysis to a dynamic problem since a bank’s risk

management method cannot be expressed properly only by solving the static optimization

problem. In fact, stock prices and interest rates vary over time and risk hedging behaviors

should be taken accordingly. It is necessary to deepen the analysis to know how the results of

our analysis would be changed by factoring in these behaviors. It is also necessary to pay

attention to the sample period for measuring the return and risk of assets.

16

References

Black, F. and Scholes, M., 1973, "The pricing of options and corporate liabilities," Journal

of Political Economics, Vol. 81, pp. 637-659.

Fischer, T. and Roehrl, A., 2003, "Risk and performance optimization for portfolios of

bonds and stocks," working paper.

Konno, H. and Kobayashi, K., 1997, "An integrated stock-bond portfolio optimization

model," Journal of Economic Dynamics and Control, Vol. 21, pp. 1427-1444.

Markowitz. H., 1959, Portfolio Selection: Efficient Diversification of Investments, Wiley, New

York.

Vasicek, O., 1977, "An equilibrium characterization of the term structure," Journal of

Financial Economics, Vol. 5, pp. 177-188.

Financial System Report, Bank of Japan, April 2012

.

17

Addendum.Derivation of Optimal Stockholding / Bondholding Ratio

1.Optimization problem

Given equation (3), equation (1) is:

max{𝑤2}

𝑤2 𝐄0 𝑆𝑇𝑆0 − 𝐄0

𝐵𝑇,𝐷 + 𝑐0,𝐷𝑇

𝐵0,𝐷 + 𝐄0

𝐵𝑇,𝐷 + 𝑐0,𝐷𝑇

𝐵0,𝐷 − 1 , (22)

Therefore, if the expected rate of return of the stock portfolio exceeds that of the bond

portfolio, that is, if the value in the brackets of the above equation is positive, from

conditional equation (2), the solution of equation (22) is the larger of the solutions of the

second-degree equation (24) as described below. If the variances of 𝐵𝑇,𝐷/𝐵0,𝐷 and 𝑆𝑇/𝑆0

are expressed as 𝑎 and 𝑐 respectively, and the covariance as 𝑏, then

𝐕0 𝜙𝑡 = 𝒘𝑇 𝑎 𝑏𝑏 𝑐

𝒘 = 𝛾 (23)

⟺ 𝑎 − 2𝑏 + 𝑐 𝑤22 + 2 𝑏 − 𝑎 𝑤2 + 𝑎 − 𝛾 = 0. (24)

Since 𝑎 − 2𝑏 + 𝑐 > 0, the solution of the above equation is equation (8).

The value in the brackets of equation (22) calculated from the parameters set in this paper

is positive mainly because the stock dividends exceed the interest rate of 3-year government

bonds13

.

2.Variance and Covariance of Stock and Bond Subportfolios

Next, the variances 𝑎 and 𝑐 and the covariance 𝑏 of the value of the bond and stock

subportfolios, 𝐵𝑡,𝐷/𝐵0,𝐷, respectively, are obtained.

First, the variance of the bond portfolio 𝑎 is derived. From equation (5):

𝑟𝑡 = 𝑒−𝜅𝑡 𝑟0 − 𝜃 + 𝜃 + 𝜍𝑟 𝑒𝜅 𝑠−𝑡 𝑡

0

𝑑𝑊𝑠(2).

If the values from 0 to 𝑡 are integrated with respect to time:

13

This paper assumes that the expected rate of return on stock prices excluding dividends is

zero. If the expected rate of return on stocks is negative, the condition may not be satisfied. In

such a case, the optimal stockholding ratio may be zero or negative (it is optimal to take short

positions on stocks).

18

𝑟𝑠

𝑡

0

𝑑𝑠 =1 − 𝑒−𝜅𝑡

𝜅 𝑟0 − 𝜃 + 𝜃𝑡 −

𝜍𝑟𝜅 𝑒−𝜅 𝑡−𝑢 − 1 𝑑𝑊𝑢

2 (25)𝑡

0

On the other hand, from equations (5) and (6):

𝑑ln𝐵𝑡,𝐷 = −𝐷𝜅 𝜃 − 𝑟𝑡 𝑑𝑡 − 𝐷𝜍𝑟𝑑𝑊𝑡(2) −

1

2𝐷2𝜍𝑟

2𝑑𝑡.

If the values from 0 to 𝑡 are integrated with respect to time:

ln𝐵𝑡,𝐷

𝐵0,𝐷= −𝐷𝜅𝜃𝑡 −

1

2𝐷2𝜍𝑟

2𝑡 + 𝐷𝜅 𝑟𝑠

𝑡

0

𝑑𝑠 − 𝐷𝜍𝑟 𝑑𝑊𝑠(2).

𝑡

0

Substituting into equation (25) and rearranging:

𝐵𝑡 ,𝐷

𝐵0,𝐷= exp −

1

2𝐷2𝜍𝑟

2𝑡 − 𝐷 𝑒−𝜅𝑡 − 1 (𝑟0 − 𝜃) − 𝐷𝜍𝑟 𝑒−𝜅(𝑡−𝑠)𝑑𝑊𝑠(2)

𝑡

0

= 𝑋 ∙ exp −𝐷𝜍𝑟 𝑒−𝜅 𝑡−𝑠 𝑑𝑊𝑠(2)

𝑡

0

. (26)

𝑋 is defined by equation (12) (Here and hereafter we describe the investment horizon 𝑇

as 𝑡). If 𝑌, which is defined by equation (13) is used, then:

𝐄0 exp −𝐷𝜍𝑟 𝑒−𝜅 𝑡−𝑠 𝑑𝑊𝑠 2

𝑡

0

= exp 𝐷2𝜍𝑟

2

4𝜅 1 − 𝑒−2𝜅𝑡 = 𝑌,

so the mean of the bond portfolio as the expected value of equation (26) is:

𝐄0 𝐵𝑡,𝐷

𝐵0,𝐷 = 𝑋𝑌, (27)

Since

𝐄0 𝐵𝑡,𝐷

𝐵0,𝐷

2

= 𝑋2𝐄0 exp −2𝐷𝜍𝑟 𝑒−𝜅 𝑡−𝑠 𝑑𝑊𝑠(2)

𝑡

0

= 𝑋2𝑌4 , (28)

from equations (27) and (28), the variance of the bond portfolio is:

𝑎 = 𝐕0 𝐵𝑡,𝐷

𝐵0,𝐷 = 𝑋2𝑌2 𝑌2 − 1 , (29)

Now, we derive the variance of the stock portfolio 𝑐. From equation (4):

19

𝑆𝑡𝑆0

= exp 𝜇𝑡 −𝜍𝑠

2𝑡

2 + 𝜍𝑠 𝑑𝑊𝑠

1 𝑡

0

, (30)

so

𝐄0 𝑆𝑡𝑆0 = 𝑒𝜇𝑡 , 𝐄0

𝑆𝑡𝑆0

2

= 𝑒 2𝜇𝑡+𝜍𝑠2𝑡 , (31)

Therefore, 𝑐 is:

𝑐 = 𝐕0 𝑆𝑡𝑆0 = 𝑒2𝜇𝑡 𝑒𝜍𝑠

2𝑡 − 1 , (32)

Lastly, we derive the covariance 𝑏. From equations (26) and (30):

𝐄0 𝐵𝑡,𝐷

𝐵0,𝐷∙𝑆𝑡𝑆0 =∙ exp 𝜇𝑡 −

𝜍𝑠2𝑡

2 𝐄0 exp −𝐷𝜍𝑟 𝑒−𝜅 𝑡−𝑠 𝑑𝑊𝑠

2 𝑡

0

+ 𝜍𝑆 𝑑𝑊𝑠 1

𝑡

0

. (33)

where, if

𝛼𝑡 = −𝐷𝜍𝑟 𝑒−𝜅 𝑡−𝑠 𝑑𝑊𝑠 2

𝑡

0

, 𝛽𝑡 = 𝜍𝑠 𝑑𝑊𝑠 1

𝑡

0

,

then:

𝐄0 𝛼𝑡 = 0, 𝐄0 𝛽𝑡 = 0, 𝐕0 𝛼𝑡 =𝐷2𝜍𝑟

2

2𝜅 1 − 𝑒−2𝜅𝑡 , 𝐕0 𝛽𝑡 = 𝜍𝑠

2𝑡

Since it is calculated as:

𝐂𝐨𝐯0 𝛼𝑡 , 𝛽𝑡 = −𝐷𝜌𝜍𝑟𝜍𝑠 1 − 𝑒−𝜅𝑡

𝜅 ,

𝐄0 𝐵𝑡,𝐷

𝐵0,𝐷∙𝑆𝑡𝑆0 = 𝑋 ∙ exp 𝜇𝑡 −

𝜍𝑠2𝑡

2 exp 𝐄0 𝛼𝑡 + 𝛽𝑡 +

1

2𝐕0 𝛼𝑡 +

1

2𝐕0 𝛽𝑡 +𝐂𝐨𝐯0 𝛼𝑡 ,𝛽𝑡

= 𝑋𝑌𝑒𝜇𝑡 exp −𝐷𝜌𝜍𝑟𝜍𝑆1 − 𝑒−𝜅𝑡

𝜅 , (34)

20

From equations (27), (31) and (34):

𝑏 = 𝐂𝐨𝐯0 𝐵𝑡 ,𝐷

𝐵0,𝐷,𝑆𝑡𝑆0 = 𝑋𝑌𝑒𝜇𝑡 exp −𝐷𝜌𝜍𝑟𝜍𝑠

1 − 𝑒−𝜅𝑡

𝜅 − 1 . (35)

(Figure 1)

Outstanding Amount of Stockholdings

(a) Major banks

(b) Regional banks

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

10

20

30

40

00 01 02 03 04 05 06 07 08 09 10 1H11

Amount of stockholdingsAmount of stockholdings/Tier1 (rhs)

(trillion yen)

Fiscal year

Multiple

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0

2

4

6

8

00 01 02 03 04 05 06 07 08 09 10 1H11

(trillion yen)

Fiscal year

Multiple

Note: excluding subsidiaries and affiliates.Source: Bank of Japan

(Figure 2)

Comprehensive Income

(a) Major banks

(b) Regional banks

-8

-6

-4

-2

0

2

4

6

8

01 02 03 04 05 06 07 08 09 10

Overall gains/losses on stockholdingsOthersComprehensive income

(trillion yen)

Fiscal year

-4

-3

-2

-1

0

1

2

3

4

01 02 03 04 05 06 07 08 09 10

Overall gains/losses on stockholdingsOthersComprehensive income

(trillion yen)

Fiscal year

Note: Overall gains/losses on stockholdings are the sum of realized gains/losses multiplied by 0.6 and changes in unrealized gains/losses on stockholdings.

Source: Bank of Japan

(Figure 3)

Correlation between Stock Prices and Interest Rates in Japan

(a) Stock prices and interest rates

(b) Correlation coefficient between stock prices and interest rates

0

1

2

3

4

5

6

7

8

9

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 120

500

1,000

1,500

2,000

2,500

3,000

3,500

TOPIX (rhs)10-year JGB yield

% pt

Years

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 Years

Average level since 2000 (0.33)

Minimum value immediately after the collapse of the bubble economy (0.63)

Note: Correlation coefficients are calculated based on daily returns on TOPIX and daily changes in 10-year government bond interest rates durning a 130-day rolling window.

Source: Bloomberg

(Figure 4)

Correlation between Stock Prices and Interest Rates in Italy

(a) Stock prices and interest rates

(b) Correlation coefficient between stock prices and interest rates

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

04 05 06 07 08 09 10 11 Years

0

1

2

3

4

5

6

7

8

03 04 05 06 07 08 09 10 110

5

10

15

20

25

30

35

40

45

50Italian stock price index (rhs)

10-year Italian government bond yield

%

Years

Note: Correlation coefficients were calculated based on daily returns on the Italian stock price index and daily changes in 10-year government bond interest rates during a 130-day rolling window.

Source: Bloomberg

(Figure 5)

Variance of Portfolio and Optimal Stockholding Ratio

(a) Major banks

(b) Regional banks

Note 1: Duration D of the bond portfolio is the actual value for major and regional banks as of the end of FY2010.Note 2: The current level of stockholdings is the actual figure as of the end of the first half of FY2011 .

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10Stockholding ratio, %

Current level of stockholdings

Variance (Volatility), %

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10Stockholding ratio, %

Current level of stockholdings

Variance (Volatility), %

(Figure 6)

Amount of Losses (Risk) when a 1% Shock Occurs and Optimal Stockholding Ratio

(a) Major banks

(b) Regional banks

Note 1: Duration D of the bond portfolio is the actual value for major and regional banks as of the end of FY2010.Note 2: The current level of Stockholding ratio is the actual figure as of the end of the first half of FY2011.

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8 9 10

Stockholding ratio , %

Amoutn of risk (trillion yen)

Current levelof stockholdings

0

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 10Stockholding ratio , %

Amount of risk (trillion yen)

Current level of stockholdings

(Figure 7)

Changes in Correlation Coefficients and Amount of Risk (Major Banks)

0

2

4

6

8

10

12

-1-0.8-0.6-0.4-0.200.20.40.60.81

Correlation coefficient

Amount of risk (trillion yen)

7.3 trillion yen

ρ=-0.63ρ=0.33

6.3 trillion yen

ρ=0

5.7 trillion yen

Note: The vertical lines, from left to right, represent the average level since 2000 ( = 0.33), no correlation ( = 0) and the realized value after the collapse of the bubble economy ( = −0.63), respectively.

(Figure 8)

Changes in Correlation Coefficients and Amount of Risk(Regional Banks)

0

1

2

3

4

5

-1-0.8-0.6-0.4-0.200.20.40.60.81

Correlation coefficient

Amount of risk (trillion yen)

3.6 trillion yen

2.4 trillion yen

ρ=-0.63ρ=0.33

2.9 trillion yen

ρ=0

Note: The vertical lines, from left to right, represent the average level since 2000 ( = 0.33), no correlation ( = 0) and the realized value after the collapse of the bubble economy ( = −0.63), respectively.

(Figure 9)

Changes in Stock Price Volatility and Amount of Risk (Major Banks)

Note: The vertical lines, from left to right, represent estimates based on the data since 2000 ( = 0.23) and estimates based on the data for 1 year including the Lehman Shock ( = 0.42), respectively.

0

2

4

6

8

10

12

0.1 0.2 0.3 0.4 0.5

Stock price volatility

Amount of risk (trillion yen)

5.7 trillion yen

10.9 trillion yen

σs=0.23 σs=0.42

(Figure 10)

Changes in Stock Price Volatility and Amount of Risk (Regional Banks)

0

1

2

3

4

5

0.1 0.2 0.3 0.4 0.5Stock price volatility

Amount of risk (trillion yen)

2.4 trillion yen

4.4 trillion yen

σs=0.23 σs=0.42

Note: The vertical lines, from left to right, represent estimates based on the data since 2000 ( = 0.23) and estimates based on the data for 1 year including the Lehman Shock ( = 0.42), respectively.

(Figure 11)

Changes in Interest Rate Volatility and Amount of Risk (Major Banks)

Note: The vertical lines, from left to right, represent estimates based on the data since 2000 ( = 0.30%) and estimates based on the data for 1 year including the Lehman Shock ( = 0.49%), respectively.

0

2

4

6

8

10

12

0.2 0.3 0.4 0.5 0.6

Interest rate volatility, %

Amount of risk (trillion yen)

5.7 trillion yen 5.8 trillion yen

σr=0.30% σr=0.49%

(Figure 12)

Changes in Interest Rate Volatility and Amount of Risk (Regional Banks)

Note: The vertical lines, from left to right, represent estimates based on the data since 2000 ( = 0.30%) and estimates based on the data for 1 year including the Lehman Shock ( = 0.49%), respectively.

0

1

2

3

4

5

0.2 0.3 0.4 0.5 0.6

Interest rate volatility, %

Amount of risk (trillion yen)

2.4 trillion yen

2.9 trillion yen

σr=0.30% σs=0.49%

(Figure 13)

Comparison of Optimal Stockholding Ratio and Current Level of Individual Banks (In a Normal Market Environment)

0

5

10

15

20

25

30

Un-feasible

Lessthan

-5

-5~ -4~ -3~ -2~ -1~ 0~ 5~ 10~

Regional banksMajor banks

Banks

%

Banks which need to reduce the stockholding ratioin order to restrain the securities portfolio risk within the capital buffer

Note: The banks which hold a positive capital buffer are listed. “Unfeasible” refers to a bank which cannot restrain the securities portfolio risk within the capital buffer despite any change in the stockholding ratio.

(Figure 14)

Comparison of Optimal Stockholding Ratio and Current Level of Individual Banks (Reversal of Correlation)

0

5

10

15

20

25

30

Un-feasible

Lessthan-5

-5~ -4~ -3~ -2~ -1~ 0~ 5~ 10~

Regional baknsMajor banks

Banks

%

Banks which need to reduce the stockholding ratioin order to restrain the securities portfolio risk within the capital buffer

Note: The banks which hold a positive capital buffer are listed. “Unfeasible” refers to a bank which cannot restrain the securities portfolio risk within the capital buffer despite any change in the stockholding ratio.

(Figure 15)

Comparison of Optimal Stockholding Ratio and Current Level of Individual Banks (a Rise of Stock Price Volatility)

0

5

10

15

20

25

30

Un-feasible

Lessthan-5

-5~ -4~ -3~ -2~ -1~ 0~ 5~ 10~

Regional banksMajor banks

Banks

%

Banks which need to reduce the stockholding ratioin order to restrain the securities portfolio risk within the capital buffer

Note: The banks which hold a positive capital buffer are listed. “Unfeasible” refers to a bank which cannot restrain the securities portfolio risk within the capital buffer despite any change in the stockholding ratio.

(Figure 16)

Comparison of Optimal Stockholding Ratio and Current Level of Individual Banks (a Rise of Interest Rate Volatility)

0

5

10

15

20

25

30

35

Un-feasible

Lessthan-5

-5~ -4~ -3~ -2~ -1~ 0~ 5~ 10~

Regional banksMajor banks

Banks

%

Banks which need to reduce the stockholding ratioin order to restrain the securities portfolio risk within the capital buffer

Note: The banks which hold a positive capital buffer are listed. “Unfeasible” refers to a bank which cannot restrain the securities portfolio risk within the capital buffer despite any change in the stockholding ratio.

(Table 1)

Basic Statistics

Mean Median Maximum Minimum StandardDeviation Skewness Kurtosis

Number ofObservation

sStock price 1,366 1,270 2,125 847 340 0.59 -0.91 2,922

Stock return -0.02 0.02 12.86 -10.01 1.46 -0.34 6.26 2,922

Interest rate 0.45 0.31 1.24 0.08 0.29 0.76 -0.78 2,945

Change in interest rates 0.00 0.00 0.14 -0.12 0.02 0.35 4.99 2,945

Note: The stock return and the interest rate in unit of % and the change in interest rates in percentage points. The stock return excludes dividends.

(Table 2)

Parameter Benchmark In times of stress

7.77% ―

23.1% 42.4%

0.45% ―

0.30% 0.49%

0.52 ―

0.33 -0.63

Values of Estimated Parameters